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Properties

Label 2-252-1.1-c5-0-8
Degree 22
Conductor 252252
Sign −1-1
Analytic cond. 40.416740.4167
Root an. cond. 6.357416.35741
Motivic weight 55
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 16·5-s − 49·7-s − 8·11-s + 684·13-s + 2.21e3·17-s − 2.69e3·19-s − 3.34e3·23-s − 2.86e3·25-s + 3.25e3·29-s + 4.78e3·31-s + 784·35-s − 1.14e4·37-s − 1.33e4·41-s − 928·43-s − 1.21e3·47-s + 2.40e3·49-s − 1.31e4·53-s + 128·55-s − 3.47e4·59-s − 1.03e3·61-s − 1.09e4·65-s + 1.01e4·67-s − 6.27e4·71-s − 1.89e4·73-s + 392·77-s + 1.14e4·79-s − 8.89e4·83-s + ⋯
L(s)  = 1  − 0.286·5-s − 0.377·7-s − 0.0199·11-s + 1.12·13-s + 1.86·17-s − 1.71·19-s − 1.31·23-s − 0.918·25-s + 0.718·29-s + 0.894·31-s + 0.108·35-s − 1.37·37-s − 1.24·41-s − 0.0765·43-s − 0.0800·47-s + 1/7·49-s − 0.641·53-s + 0.00570·55-s − 1.29·59-s − 0.0355·61-s − 0.321·65-s + 0.275·67-s − 1.47·71-s − 0.415·73-s + 0.00753·77-s + 0.205·79-s − 1.41·83-s + ⋯

Functional equation

Λ(s)=(252s/2 ΓC(s) L(s)=(− Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}
Λ(s)=(252s/2 ΓC(s+5/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 252252    =    22⋅32⋅72^{2} \cdot 3^{2} \cdot 7
Sign: −1-1
Analytic conductor: 40.416740.4167
Root analytic conductor: 6.357416.35741
Motivic weight: 55
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 252, ( :5/2), −1)(2,\ 252,\ (\ :5/2),\ -1)

Particular Values

L(3)L(3) == 00
L(12)L(\frac12) == 00
L(72)L(\frac{7}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
3 1 1
7 1+p2T 1 + p^{2} T
good5 1+16T+p5T2 1 + 16 T + p^{5} T^{2}
11 1+8T+p5T2 1 + 8 T + p^{5} T^{2}
13 1−684T+p5T2 1 - 684 T + p^{5} T^{2}
17 1−2218T+p5T2 1 - 2218 T + p^{5} T^{2}
19 1+142pT+p5T2 1 + 142 p T + p^{5} T^{2}
23 1+3344T+p5T2 1 + 3344 T + p^{5} T^{2}
29 1−3254T+p5T2 1 - 3254 T + p^{5} T^{2}
31 1−4788T+p5T2 1 - 4788 T + p^{5} T^{2}
37 1+310pT+p5T2 1 + 310 p T + p^{5} T^{2}
41 1+13350T+p5T2 1 + 13350 T + p^{5} T^{2}
43 1+928T+p5T2 1 + 928 T + p^{5} T^{2}
47 1+1212T+p5T2 1 + 1212 T + p^{5} T^{2}
53 1+13110T+p5T2 1 + 13110 T + p^{5} T^{2}
59 1+34702T+p5T2 1 + 34702 T + p^{5} T^{2}
61 1+1032T+p5T2 1 + 1032 T + p^{5} T^{2}
67 1−10108T+p5T2 1 - 10108 T + p^{5} T^{2}
71 1+62720T+p5T2 1 + 62720 T + p^{5} T^{2}
73 1+18926T+p5T2 1 + 18926 T + p^{5} T^{2}
79 1−11400T+p5T2 1 - 11400 T + p^{5} T^{2}
83 1+88958T+p5T2 1 + 88958 T + p^{5} T^{2}
89 1+19722T+p5T2 1 + 19722 T + p^{5} T^{2}
97 1−17062T+p5T2 1 - 17062 T + p^{5} T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−10.55459690028161290337774082771, −9.937316717899621250490684495683, −8.575112552914830418078523997200, −7.929367140853306396169152744262, −6.55311814864751536479426054348, −5.73197414798066843777944272257, −4.21717783589933081406849014672, −3.23641400235369632726176733783, −1.58117296822369444835957940989, 0, 1.58117296822369444835957940989, 3.23641400235369632726176733783, 4.21717783589933081406849014672, 5.73197414798066843777944272257, 6.55311814864751536479426054348, 7.929367140853306396169152744262, 8.575112552914830418078523997200, 9.937316717899621250490684495683, 10.55459690028161290337774082771

Graph of the ZZ-function along the critical line